Solitons can generally be defined as a member of a class of solutions to nonlinear equations or non linear propagation problems. Such solutions can be characterized by certain amplitude or power levels and certain pulse shapes that can be interrelated, where the solutions can propagate with an unchanging pulseshape over an indefinite distance, or can display a slow periodic oscillation with distance through a set of recurring characteristic pulseshapes. Depending on the particular nonlinear equation, the soliton pulses can have different shapes and the velocities of propagation and the distances for periodic recurrence generally depend on both the nonlinear equation and the pulse amplitude.
Since the invention of the Laser various physical phenomena have been used to confine light with light. The required processes invoke a local increase in the optical refractive index primarily due to the third or even higher order nonlinear optical response of a material. In particular, for laser beams free to diffract in two-transverse dimensions, which is our geometry, saturable third order processes have been implemented for this purpose. They suffer from the requirement that the necessary higher order response is typically available only near a resonance condition which usually is accompanied by inherent optical losses. The required large saturable nonlinearities have mainly been achieved in atomic gas systems. See J. E. Bjorkholm, A. Ashkin, "cw Self-Focusing and Self-Trapping of Light in Sodium Vapor", Phys. Rev. Lett., 32, 129-132 (1974).
When only one transverse dimension of an optical beam is allowed to diffract, the second one being confined by strong linear wave-guiding, one dimensional spatial solitons are possible and have been observed in a few dense transparent third order nonlinear optical materials where excellent planar waveguides can be fabricated. See A. Barthelemy, S. Mancur, C. Froehly, "Propagation soliton et auto-confinement de faisceaux laser par non-linearite optique de Kerr", Opt. Comm., 55, 201-206 (1985); and See J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel. D. E. Leaird, E. M. Vogel, P. W. E. Smith, "Observation of spatial optical solitons in a nonlinear glass waveguide", Opt. Lett. 15, 471-473 (1990).
Because of the transparency requirement the latter methods suffer from extreme light intensifies and are only applicable to one free transverse dimension. Indeed, mathematically in more than one dimension the equation which governs the propagation of a single field in a material with an intensity-dependent refractive index does not lead to solitary beams (similar to solitons but with different collision properties) and in general unstable filaments occur. Solving the latter problems, photorefractive solitons have successively been demonstrated in more than one dimension at milliwatt power levels. However in order to achieve the photorefractive effect electrical carriers in the material have to be created, resulting in optical losses, just like in atomic systems. See G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. D. Porto, E. J. Sharp, R. R. Neurgaonkar, "Observation of Self-Trapping of an Optical Beam Due to the Photorefractive Effect", Phys. Rev. Lett., 71, 533-536 (1993).
Furthermore, because of the necessary formation and diffusion of photocarriers the photorefractive effect exhibits an inherently slow response. Recently it has been realized that lower order nonlinearities, second order processes, in noncentrosymmetric dense optical media could lead to effects equivalent to those observed in third order nonlinear optical materials, namely self focusing and defocusing. See R. deSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. W. VanStryland, H. Vanherzeele, "Self-focusing and self-defocusing by cascaded second-order effects in KTP", Opt. Lett., 18, 28-30 (1992). One of these effects is a nonlinear distortion of the phase front of the optical beam. See G. I. Stegeman, M. Sheik-Bahae, E. W. VanStryland, G. Assanto, "Large nonlinear phase-shifts in second-order nonlinear-optical processes", Opt. Lett., 18, 13-15 (1993). Theoretically the power law dependence of the phase distortion on the input optical field led to the prediction of solitary beams of higher dimension than one. See Y. N. Karamzin, A. P. Sukhorukov, "Mutual focusing of high-power light Sov. Phys.-J.E.T.P., 41, 414-420 (1976). This includes the two-dimensional case. Full theoretical treatments indeed prove that such soliton-like beams should exist in quadratic media. See Y. N. Karamzin, A. P. Sukhomkov, "Mutual focusing of high-power light Sov.Phys.-J.E.T.P., 41, 414-420 (1976); and See K. Hayata, M. Koshiba, "Multidimensional Solitons in Quadratic Nonlinear Media", Phys. Rev. Lett., 71, 3275-3278 (1993); and See L. Torner, C.R. Menyuk, W. E. Torruellas, G. I. Stegeman, "Two-dimensional solitons with second-order nonlinearities", Opt. Lett., 20, 13-15 (1995).
In the so called "cascading" scheme, second harmonic is generated in the crystal. When this harmonic field mixes back with the fundamental field, the down-converted fields interfere with the fundamental which was not converted to second harmonic, resulting in optical phase distortions in all of the fields linked by the process. This approach can lead to effective third order nonlinearities larger than those observed in intrinsic third order materials. But one should note that in this case the local refractive index remains unchanged, only a distortion of the optical phase front is produced. The nonlinear phase distortion intrinsically saturates when the three fields involved are strongly coupled via the second order polarization, and is proportional to the field amplitudes and not their intensities.
Various U.S. patents have been granted generally along the line of generating time dependent solitary waves along superconductors and optical fibers. For example, U.S. Pat. Nos. 4,361,768 to Rajeevakumar; 5,134,621 to Marshall; 5,157,744 to Korotky; 5,305,006 to Lee; and 5,381,397 to Okada describes various two dimensional time dependent soliton generation systems for superconductors. U.S. Pat. Nos. 4,635,263 to Mollenauer; 4,881,788 to Doran; 5,305,336 to Adar; 5,140,656 to Hasegawa et al.; 5,191,628 to Byron; 5,195,160 to Byron; 5,201,017 to Byron; 5,357,364 to Gordon et at.; 5,363,386 to Smith describes soliton generation along optical fiber that are also only time dependent.